Curve Fitting in Excel: A Tutorial on Fitting a Complex Nonlinear Regression Model to Your Data

Introduction

Welcome to our comprehensive tutorial on Excel curve fitting. Whether you're a student, researcher, or professional, mastering the art of curve fitting in Excel can transform your data analysis. This guide will walk you through the simple yet effective techniques to apply the best fit curve to your datasets using nonlinear regression.

In today's data-driven world, the ability to analyze and interpret data is essential. One powerful tool that can help you achieve this is curve fitting in Excel. Excel can be effectively utilized as a curve fitting tool, much like other curve fitting software such as ndCurveMaster.

Whether you're a beginner or have some experience with Excel, this tutorial will guide you through mastering curve fit techniques in Excel. You'll learn how to use Excel's built-in functions and tools to fit curves to your data, enabling you to make more accurate predictions and gain deeper insights. In this tutorial, I will demonstrate how to fit curves to data that encompass up to three independent variables.

By the end of this tutorial, you'll have a solid foundation in curve fitting techniques and the confidence to apply them to your own data sets. Whether you're a business analyst, a researcher, or simply someone looking to enhance their Excel skills, get ready to dive into the world of curve fitting in Excel.

Why Curve Fitting is Essential

Curve fitting is a fundamental technique in data analysis, offering insights that are critical for both research and practical applications. By fitting curves to a set of data points, we can model relationships between variables, predict future trends, and understand underlying patterns in complex datasets. This process is not just about connecting dots; it's about uncovering the story your data tells.

Unveiling the Patterns in Data

At its core, curve fitting helps in revealing the patterns hidden within the data. Whether it's a linear relationship that suggests a direct correlation between variables or a nonlinear one that uncovers more complex interactions, curve fitting provides a visual and mathematical representation of these relationships. This is essential in fields like physics, chemistry, biology, economics, and engineering, where understanding the dynamics between variables can lead to groundbreaking discoveries and innovations.

Enhancing Decision Making

In the business world, curve fitting plays a pivotal role in decision-making processes. By analyzing historical data and trends, companies can forecast future outcomes, such as sales trends, stock market movements, or consumer behavior changes. This predictive power enables businesses to make informed decisions, allocate resources more efficiently, and stay ahead in competitive markets.

Improving Accuracy of Predictions

Curve fitting also significantly enhances the accuracy of predictions. By selecting the most appropriate model that fits the data, analysts can minimize errors and increase the reliability of their forecasts. This is crucial in scientific research, where precise predictions can determine the success of experiments and validity of theories.

Simplifying Complex Relationships

Data often involves multiple variables with intricate relationships that are difficult to analyze using simple linear models. Curve fitting allows for the application of nonlinear models that can accurately capture these complexities, providing a clearer understanding of how variables interact with each other in real-world scenarios.

Facilitating Data Visualization

Visual representation is a powerful tool for data analysis, and curve fitting contributes to creating more meaningful and interpretable visualizations. By illustrating the trends and patterns through curves, it becomes easier for analysts, stakeholders, and non-experts to grasp the insights derived from data analysis, making complex information accessible to a wider audience.

In conclusion, curve fitting is an essential tool in the data analyst's arsenal. It bridges the gap between raw data and actionable insights, enabling professionals across various disciplines to unlock the full potential of their data. Whether it's for scientific discovery, business strategy, or understanding complex systems, curve fitting provides the foundation for a deeper understanding and more accurate predictions.

Simple Exponential, Polynomial or Logarithmic Curve Fitting Using Excel and Trendline Option

To fit a function to your data in Excel, you can simply utilize the trendline feature. However, this approach is applicable only when you aim to find a function based on a single input variable, in other words, a straightforward model like:

To get started, select the two columns of data points, create a scatterplot, and then mark the points on the chart. Next, use the context menu to choose the Add trendline option. After selecting this option, you can format trendline and fit it to your data using various functions, including exponential, linear, logarithmic, polynomial, power, and moving average. Following this procedure, a trendline will be added to your chart, and you can even display equation on chart along with the R-squared value.

This method, while simple to grasp, often comes up short due to its limited selection of available functions and, most notably, its inability to handle functions involving multiple variables. It's important to mention that this approach is commonly used and extensively covered in many online resources, which is why I won't be discussing it further in this article.

In the next part of this article, I'll show you a tutorial on how to use Excel for fitting functions to data when the data comprises multiple independent variables (inputs), for example:

Steps to Curve Fit in Excel

Defining the Regression Equation and Variables

In this tutorial, we will fit the coefficients a0, a1, a2, a3, and the exponents b1, b2, b3 of the following nonlinear regression equation to data comprising 17 measurements of the Y variable and three independent variables, x1, x2, and x3:

Y = a0 + a1*x1^b1 + a2*x2^b2 + a3*x3^b3 (1)

Typically, all variables are measurement results when fitting curves to data. However, in this tutorial, the independent variable Y is determined using the exponential equation:

Y = 3 + 3*x1^2.5 + 4*x2^3 - 3.5*x3^0.5 (2)

Knowing the true relationship between variables will make it possible to assess the reliability of the discovered equations in the curve fitting process.

To begin, create three columns in your Excel spreadsheet to house the variables x1, x2, and x3:

Next, create a column for Y and calculate its values using equation (2):

Y = 3 + 3*x1^2.5 + 4*x2^3 - 3.5*x3^0.5

Here's the formula for the first row:

Implementing the Best Fit Curve in Excel

In this section, curve fitting will be parameterized based on the regression coefficients and exponents. Start by adding cells with values for coefficients a0, a1, a2, a3, and exponents b1, b2, b3. Initially, assign them the value "1":

Create an additional column labeled Y*. This column will be calculated based on x1, x2, x3, coefficients a0 - a3, and exponents b1 - b2, using the formula (3). The following image displays a column with Y* values:

Implementing Least Squares for Finding the Best Fit

To measure the accuracy of our estimation, we will employ the Root Mean Square Error (RMSE), calculated as follows:

RMSE = √(Σ(Y(i) - Y*(i))^2/n) (3)

RMSE quantifies the differences between actual data (variable Y) and the estimated values (Y*). A lower RMSE indicates a better fit. In our tutorial, RMSE is used as the criterion for curve fitting.

Create another column, (Y - Y* )^2, to calculate the squares of the differences between Y and Y* using the formula: (D2 - E2)^2, as shown below:

Subsequently, calculate the sum of all the differences (in column F), divide this sum by the number of elements (n=16), and finally, calculate the square root of this number using formula (3).

This gives you the RMSE error. The following figure displays the sheet with the formula:

Implementing the Pearson Correlation Coefficient to Assess Fit Accuracy

In the realm of curve fitting and data analysis, accurately assessing the quality of fit is paramount. One of the most effective tools for this purpose is the Pearson Correlation Coefficient (PCC). This statistical measure evaluates the linear correlation between two variables, providing insights into the strength and direction of their linear relationship. By implementing the PCC, analysts can quantitatively determine how well a curve fits a set of data points, ensuring the selected model accurately reflects the underlying trends.

The Pearson Correlation Coefficient, ranging from -1 to 1, offers a clear metric for correlation strength. A value close to 1 indicates a strong positive correlation, meaning as one variable increases, the other does too. Conversely, a value near -1 signifies a strong negative correlation, where one variable's increase corresponds to the other's decrease. A coefficient around 0 suggests no linear correlation, highlighting the need for a different model or analysis approach.

In curve fitting, the PCC is invaluable for comparing the observed data points to the values predicted by the curve. A high positive correlation coefficient indicates that the curve accurately represents the data trend, validating the model's effectiveness. This quantitative assessment aids in model selection, ensuring the chosen curve fitting approach captures the essential dynamics of the data set.

Excel offers a built-in function to calculate Pearson's correlation coefficient. You can add it using the formula:

=PEARSON(D2:D17, E2:E17)

Utilizing Scatter Plot Charts to Evaluate Fit Accuracy

A scatter plot is a crucial tool for assessing how well a model fits the data. It offers a visual way to examine the connection between the variable Y and its predicted value Y*. By plotting these points on a graph, you can quickly observe how well the model's predictions align with the actual data. A strong alignment along a diagonal line (Y = Y*) in the scatter plot indicates a good model fit, while deviations from this line reveal where the model may overestimate or underestimate values. The scatter plot is a crucial tool for evaluating the accuracy and effectiveness of a predictive model.

This chart visualizes how well the model fits the data. The more closely the points align, running along a diagonal line inclined at a 45-degree angle to the x-axis, the better the fit.

Using Solver for Nonlinear Curve Fitting

Now, we will proceed with the curve fitting process. This will be a multidimensional task since it is based on three independent variables and involves nonlinear regression.

To start, add the Solver add-in to Excel. Under the "Excel Options" tab, navigate to the "Add-ins" tab, select "Solver Add-in," click "Go," and choose the "Solver" option:

If the installation is successful, you will see the "Solver" button on the "Data" tab, in the "Analysis" section:

Once you have the "Solver" add-on installed, you can begin curve fitting. Click on the "Solver" button and a window will appear:

In the "Set Objective" field, select cell $F$18 - this cell contains the RMSE values:

In the "To" option, choose "Min" because Solver will seek a solution that minimizes RMSE:

Click on the "By changing Variable Cells:" field and specify on the sheet the coefficients and exponents of the model with values of "1," i.e., cells: $E$19:$E$25, as shown below:

Uncheck the "Make Unconstrained Variables Non-Negative" option, allowing the values to be negative. Do not change your search method; the "GRG Nonlinear" method is suitable. Here's a window with all the settings:

To perform curve fitting in Excel, click the "Solve" button. Once you've clicked it, the Solver will calculate the coefficients and exponents of the model, minimizing the RMSE value in cell F18. Click "Solve," and a window should appear:

In the background, you can already observe changes in the coefficients and exponents of the model, and the scatter plot looks promising. Click "OK" in this window to complete the curve fitting process. Below, you'll find a spreadsheet displaying the optimal model coefficients:

By using Excel for curve fitting, we've calculated the optimal constants and exponents for our model:

• a0: 0.894179
• a1: 3.518039
• a2: 1.314086
• a3: -0.066208
• b1: 2.6458
• b2: 2.11196
• b3: -4.163233

By introducing these coefficients into the regression equation (1), we obtain the following formula:

Y = 0.89 + 3.518 * x1^2.65 + 1.314 * x2^2.11 - 0.066 * x3^-4.16

These coefficients are crucial in understanding the relationship between our variables and fitting the curve effectively.

You can download the spreadsheet illustrating the above solution from the link: Download the tutorial spreadsheet.

Results: Formula Accuracy Assessment

How accurate is the solution? This will be revealed by RMSE, Pearson's correlation coefficient, and the scatter plot. The RMSE error is small, at 4.38, and the Pearson correlation coefficient is high, at 0.99999715. As evident in the picture above, the points on the scatter plot align closely to a straight line inclined at a 45-degree angle. This suggests a very good fit.

Additionally, when comparing the obtained equation to formula (2), which was used to calculate the Y values, it becomes evident that some of the coefficients and exponents closely resemble those in equation (2). For instance, the coefficient a1 and exponents b1 and b2.

Curve Fitting with ndCurveMaster: Tips for Achieving Accurate Fits

The challenge we tackled demanded four-dimensional curve fitting. As you've seen, using Excel for this purpose delivered some pretty decent results. Now, let's stack these results up against those achieved with ndCurveMaster, a top-notch curve fitting tool designed for fitting functions with any number of variables.

You can download a trial version of the program for Windows or MAC from the following link: ndCurveMaster Download

To start, you'll need to create a proper data file. Begin by copying the values of the variables x1, x2, x3, and Y to a separate sheet, and then save this sheet to disk. Your data sheet should look something like this:

Next, open ndCurveMaster, click the "Open" button, and open a data file. You can download the data file from the following link: Download the tutorial spreadsheet.

After the file has loaded, pick the right sheet in the document:

Next, in the "Input Data" window, select the variables Y, x1, x2, and x3, as shown below:

Press OK. When the main window of the program with the Multiple Linear Regression model appears:

Click on the "Advanced Search" button. This action will kickstart a randomized search, simultaneously employing iterative searching to discover the best model via nonlinear regression.

After 15 seconds, you will receive several solutions, and the best one is:

Y = 3 + 3 · x1^2.5 + 4 · x2^3 + (-3.5) · x3^(1/2)

For this solution, RMSE equals 0, and Pearson's correlation coefficient is 1.

As you can see, the coefficients and exponents in the equation found by ndCurveMaster align perfectly with those employed to calculate variable Y (formula (2)).

Conclusion

Using Excel for curve fitting can give you pretty decent results. Let's take a look at a comparison between Excel and the specialized multi-dimensional curve fitting program, ndCurveMaster. The results in the table show that ndCurveMaster hit the bullseye with 100% accuracy, outshining Excel by a wide margin and practically revealing the data's underlying equation. While Excel didn't quite crack the code for variable Y, ndCurveMaster nailed it. Nonetheless, the solution you get using Excel is still quite impressive, especially when you consider it's a DIY tool.

Mastering curve fitting in Excel is just the beginning. Explore our resources for more advanced techniques, or enhance your data analysis toolkit with ndCurveMaster. Start your journey towards data mastery today.